EM theory and its application to microwave remote sensing

1. EM theory and its application to microwave remote sensing Chris Allen (callen@eecs.ku.edu) Course website URL people.eecs.ku.edu/~callen/823/EECS823.htm

2. Outline • Plane wave propagation • Lossless media • Lossy media • Polarization and coherence • Fresnel reflection and transmission • Layered media • EM spectra, bands, and energy sources

3. Plane wave propagation • Plane wave propagation through lossless and lossy media is fundamental to microwave remote sensing. • Consider the wave equation and plane waves in homogeneous unbounded, lossless media • Plane waves – constant phase and amplitude in the plane • Homogeneous – electrical and magnetic parameters do not vary with throughout the medium • Beginning with Maxwell’s equations and assuming a homogeneous, source-free medium leads to the homogeneous wave equation • where • E is the electric field vector (V/m) [note that bolded symbols denote vectors] •  is the medium’s magnetic permeability (H/m) [H: Henrys] •  is the medium’s permittivity (F/m) [F: Farads]

4. Plane wave propagation • Assuming sinusoidal time dependence • where  is the radian frequency (rad/s) • r is the displacement vectorand Re{} is the real operator • E(r,t) satisfies the wave equation if • Using phasor representation (i.e., e.jt is understood) and assuming a rectangular coordinate system, the solution has the general form of • where E0 is a constant vector and

5. Plane wave propagation • A more compact form results from letting • where k is the propagation vector, and • k = |k|is called the wave number (rad/m) • resulting in • Finally reintroducing the time dependence and expressing only the real-time field component yields • This equation represents two waves propagating in opposite directions defined by the propagation vector k

6. Plane wave propagation • Rotating the Cartesian coordinate system such that the zaxis aligns with k yields • representing two waves propagating in the + z and – z directions. • The argument of the cosine function contains two phase terms: the time phase, t, and the space phase, kz. • If we use the – part of the ± solution we have • representing a wave whose constant phase moves in the positive z direction. As time increases, z must increase to maintain a constant phase argument.

7. Plane wave propagation • The time phase component is characterized by  where • f is the frequency (Hz) and T is the time period (s). • Similarly the space phase component depends on k where •  is the space period (m), or wavelength, in the medium which can also be expressed as

8. Plane wave propagation • Consider now the electric field’s phase for a positive-traveling wave, i.e., t – kz. • A surface on which this phase is constant requires • For any given time t, this surface represents a plane defined by z = constant, on which both the phase and amplitude are constant. As time progresses, this plane of constant phase and amplitude advances along the z axis, hence the name uniform plane wave. • The rate at which this plane advances along the z axisis called the phase velocity, v (m/s)

9. Plane wave propagation • Given an uniform plane E-field solution to the wave equation, the H-field is found using Maxwell’s equations • From the E-field component in the x-axis direction, Ex, is found the H-field component in the y-axis direction, Hy, as • where  is the intrinsic impedance () of the medium • Note that Ex and Hy are related through the intrinsic impedance similar to how voltage and current in a circuit are related through Ohm’s law. • Note also the orthogonality of the E, H, and k vectors.

10. Plane waves in a lossy medium • A lossy medium is characterized by its permeability, , permittivity, , and conductivity,  (S/m) [S: Siemens]. Maxwell’s equations for a source-free medium become • And the corresponding wave equation remains • where the wave number is • Note that for a lossless medium, k is purely real when  = 0 and both  and  are real

11. Plane waves in a lossy medium • For a lossy medium k is complex • due to   0 or either  or  are complex • For lossless media the imaginary parts of the permeability and permittivity are zero. • Non-zero imaginary terms ( > 0 and  > 0) represent mechanisms for converting a portion of the electromagnetic wave’s energy into heat, resulting in a loss of wave energy.

12. Plane waves in a lossy medium • Consider the complex electric field plane wave propagation along the positive z axis • whereas for the lossless case k was real, in a lossy medium k is complex and is related to the propagation factor or propagation constant,  (1/m), by • such that • where  and  are real quantities and  is the attenuation constant (Np/m) and  is the phase constant (rad/m)[Np = Neper]

13. Plane waves in a lossy medium • Clearly for a wave travelling along the +z axis • as z increases, the magnitude of the electric field decreases. • The real time expression for the x-axis field component is • The attenuation constant is the real part of jk • The phase constant is the imaginary part of of jk • Note: (Neper/m)  8.686 (dB/Neper) = (dB/m)

14. Plane waves in a lossy medium • In a lossy medium, the intrinsic impedance is also complex • giving rise to a non-zero phase relationship between the E and H field components. • While a medium’s loss may be due to its conductivity, or the imaginary components of permittivity or permeability, in most microwave remote sensing applications the magnetization loss () is negligible. • Exceptions include the ferrous-rich sands found in Hawaii and soils on the Martian surface. • Therefore the  term will be neglected from now on. • A material’s permeability is usually specified relative to that of free space, o, (o = 4  10-7 H/m), as  = r  oand typically r = 1

15. Plane waves in a lossy medium • A medium’s loss may be due to its conduction loss( > 0 but finite) or its polarization loss ( > 0). • Conduction loss is gives rise to a conduction current • where electrical energy is converted to heat energy due to ohmic losses.

16. Plane waves in a lossy medium • Polarization loss is due to a displacement current, similar to current through a capacitor. For an ideal dielectric, equal amounts of energy are stored and released during each cycle. For lossy dielectrics, some of the stored energy is converted into heat. • The imaginary part of the displacement current is in-phase from the E field, and hence contributes to real energy loss.

17. Plane waves in a lossy medium • For a sinusoidal time variation, there is a transition frequency, t = 2 ft, where these two current components are equal

18. Plane waves in a lossy medium • For dielectric materials, these two loss mechanisms are often combined into a single imaginary component as • The permittivity of dielectric materials is usually specified relative to the permittivity of free space, o, where (o = 8.854  10-12 F/m) as • where

19. Plane waves in a lossy medium • Often instead of specifying the r, a material’s loss tangent tan  where

20. Plane waves in a lossy medium • Clearly since the loss mechanism due to conductivity is frequency dependent, a medium’s loss characteristics may also be frequency dependent. • At low frequencies where the conductivity introduces signficant loss, the intrinsic impedance and phase velocity will be frequency dependent. • At frequencies where the conductive loss is dimished, the intrinsic impedance and phase velocity will become frequency independent.

21. Plane waves in a lossy medium • Low-loss media(i.e., tan   1) • For the case of low-loss media, the expressions for v, , , and  can be simplified to be • where c = 3  108 m/s • where o = 120    377 

22. Plane waves in a lossy medium • High-loss media(i.e., tan  >> 1) • For the case of high-loss media, the expressions for ,  and  can be simplified to • Also, when an electromagnetic wave impinges on a conducting medium, the field amplitude decreases exponentially with depth. • At a depth termed the skin depth,  (m), the field amplitude is e-1 of its value at the surface, where

23. Plane waves in a lossy medium • General lossy media • For media that is neither high loss nor low loss, the simplifying approximations  and  do not apply. • For these cases we have • where ko is the free-space wave numberand o is the free-space wavelength • It is sometimes useful to refer to a medium’s refractive index, n, where

24. Plane waves in a lossy medium

25. Plane waves in a lossy medium

26. Polarization of plane waves and coherence • For a +z-axis propagating uniform plane wave, the E-field components must lie in the xy plane • For any point on the xy-plane, the E-field varies with time. • The wave’s polarization is associated with the curve the tip of the E-field vector traces out. • A straight line indicates a linear polarization (i.e., y = 0 or180) with tilt angle 

27. Polarization of plane waves and coherence • A circle indicates circular polarization • where Exo = Eyoandy =  90 • Left-hand circular polarization (LCP), which results when  y= +90, has the E-field rotating once each cycle in the direction the fingers of the left hand point when gripping the z-axiswith the thumb in the +z direction. • Right-hand circular polarization (RCP), which results when  y= -90, has the E-field rotating in the same direction as the fingers when gripping the z-axis with the right hand so that the thumb is in the +z direction.

28. Polarization of plane waves and coherence • An elliptical pattern indicates elliptical polarization • where Exoand Eyo > 0 andy  0, 90, or 180yet these variables remain constant over time. • A wave is unpolarized when the amplitudeand phase relationships of the orthogonalcomponents are time varying.

29. Polarization of plane waves and coherence • Signals produced by single-frequency or multifrequency transmitters are typically completely polarized • Signals emitted by physical objects, irregular terrain, or inhomogeneous media are usually broadband and are composed of many statistically independent waves with different polarizations • If there is no correlation between the component waves of such a signal it is incoherent or unpolarized • Between these two extremes (completely polarized and unpolarized) are the partially-polarized signals that result when polarized signals are scattered by random targets • While characterization of completely polarized plane waves is fairly straightforward, the characterization of these partially-polarized signals is more challenging

30. Polarization of plane waves and coherence • An analysis technique to evaluate the state of polarization or degree of coherence of a plane wave involves the magnitude of the normalized cross-correlation of the x and y phasor components as • where * denotes the complex conjugateand denotes the average operator over some finite time interval T • For completely polarized signals xy = 1 while for incoherent or unpolarized signals xy = 0. • For partially-polarized or partially-coherent signals • 0 < xy < 1

31. Polarization of plane waves and coherence • To evaluate the degree of polarization (DOP), another measure of a signal’s polarization, involves the Stokes parameters • The DOP is found by • where DOP = 1 for a completely polarized signal, a DOP = 0 for an unpolarized signal, and for partially-polarized signals • 0 < DOP < 1

32. Polarization of plane waves and coherence • The Poincaré sphere is a tool for visualizing the continuum of polarization states. • Derived from the Stokes parameters, the sphere maps linear polarizations on the equator (LVP: linear vertical pol.; LHP: linear horizontal pol.) and the right (RCP) and left (LCP) circular polarizations at the north and south poles. • Points opposite one another on the sphere represent orthogonal polarizations. • Points representing completely-polarized signals lie on the sphere’s surface while points representing partially-polarized waves appear within the sphere.

33. Electromagnetic phenomena • Speed of light • c: speed of light in vacuum (2.99792458  108 m/s  3  108 m/s) • n: refractive index of material (n  1) • v: speed of light in material (m/s), v = c/n • Wavelength and frequency • f: frequency (Hz) • : wavelength (m) • o: wavelength in free space (m) • f: bandwidth in frequency (Hz) • : bandwidth in wavelength (m) • In vacuum, (n = 1, v = c) • In a medium, (n  1, v  c)

34. Fresnel reflection and transmission • Now consider the electromagnetic interactions as a plane wave impinges on a planar boundary between two different homogeneous media with semi-infinite extent • Properties of interest include reflection, refraction, transmission • Solutions are found by satisfying the requirement for the continuity of tangential E and H fields across the boundary • Simplifying assumptions: • • Plane wave propagation • • Smooth, planar interface infinite in extent • • Linear, isotropic refractive indices • • Semi-infinite media

35. Fresnel reflection and transmission • Snell’s law • n1: refractive index of medium 1 (incidence side) • n2: refractive index of medium 2 • i: incidence angle (measured from surface normal) • r: reflected angle (measured from surface normal) • t: transmitted angle (measured from surface normal) • Requirement for continuity of tangential E and H fields across the boundary yields • Reflected • Transmitted

36. Fresnel reflection and transmission • Fresnel equations • Predicts reflected and transmitted vector field quantities at a plane interface • Satisfies requirement for continuity of tangential E and H fields across the boundary • Consequently the interaction is polarization dependent • Arbitrarily polarized incident plane wave can be decomposed into two orthogonal linear polarizations: perpendicular () and parallel (//) • Separate solutions for perpendicular and parallel cases • Perpendicular and parallel refer to E-field orientation with respect to the plane of incidence(plane containing incident, reflected, and transmitted rays) • These same polarizations have other names as well

37. Fresnel reflection and transmission • Perpendicular (horizontal) case • Reflection coefficient (relates to field strength) (sometimes represented by , , or r). • Transmission coefficient (relates to field strength) • Note that 1 + R  = T

38. Fresnel reflection and transmission • Perpendicular (horizontal) case • Reflectivity (relates to power or intensity) • Transmissivity (relates to power or intensity) • (sometimes represented by T) • or • Note that += 1 which satisfies the conservation of energy

39. Fresnel reflection and transmission • Parallel (vertical) case • Reflection coefficient (relates to field strength) • Transmission coefficient (relates to field strength) • Note that 1 + R // = T//

40. Fresnel reflection and transmission • Parallel (vertical) case • Reflectivity (relates to power or intensity) • Transmissivity (relates to power or intensity) • or • Note that //+//= 1 which satisfies the conservation of energy

41. Fresnel reflection and transmission • Special cases • Normal incidence (i = 0) • i = r = t = 0, cos  = 1 • Brewster angle • B: angle where reflection coefficient for parallel (vertical) polarized field goes to zero • i.e., at  = B,// = 0, // = 1 (note polarization dependence)

42. Fresnel reflection and transmission • Special cases • Critical angle • C: incidence angle at which total internal reflection occurs (for n1 > n2) • i.e., at C,// =  = 1, // =  = 0 (note polarization independence) • Evanescent waves exist in medium 2, with imaginary propagation coefficients meaning they decay rapidly with distance z from the boundary.

43. Fresnel reflection and transmission • Normal incidence reflectioncoefficient for some typicalgeological contacts

44. Fresnel reflection and transmission • Example #1 • Consider the case where a plane wave impinges on a planar boundary between homogenous ice ( = 3.14, n = 1.78) and air ( = 1, n = 1). • From the formulas presented earlier • Normal reflectivity = -11.0 dB • Normal transmissivity = -0.4 dB • Critical angle, C = 34.4° • Brewster angle, B = 29.4°

45. Fresnel reflection and transmission • Fresnel reflection and transmission coefficients vs. incidence angle at an ice-air boundary (ice = 3.14, air = 1)

46. Fresnel reflection and transmission • Reflectivity and transmissivity expressed in linear units vs. incidence angle at an ice-air boundary (ice = 3.14, air = 1)

47. Fresnel reflection and transmission • Reflectivity and transmissivity expressed in decibels vs. incidence angle at an ice-air boundary (ice = 3.14, air = 1)

48. Fresnel reflection and transmission • Example #2 • Consider the case where a plane wave impinges on a planar boundary between homogenous ice ( = 3.14, n = 1.78) and rock ( = 5, n = 2.24). • From the formulas presented earlier • Normal reflectivity = -19.1 dB • Normal transmissivity = -0.1 dB • Critical angle, C = NA • Brewster angle, B = 51.6°

49. Fresnel reflection and transmission • Fresnel reflection and transmission coefficients vs. incidence angle at an ice-rock boundary (ice = 3.14, rock = 5)

50. Fresnel reflection and transmission • Reflectivity and transmissivity expressed in linear units vs. incidence angle at an ice-rock boundary (ice = 3.14, rock = 5)